Nuprl Lemma : comma-cat_wf
∀[A,B,C:SmallCategory]. ∀[S:Functor(A;C)]. ∀[T:Functor(B;C)]. ((S ↓ T) ∈ SmallCategory)
Proof
Definitions occuring in Statement :
comma-cat: (S ↓ T),
cat-functor: Functor(C1;C2),
small-category: SmallCategory,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
comma-cat: (S ↓ T),
spreadn: spread3,
pi2: snd(t),
pi1: fst(t),
prop: ℙ,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
so_lambda: so_lambda(x,y,z,w,v.t[x; y; z; w; v]),
so_apply: x[s1;s2;s3;s4;s5],
uimplies: b supposing a,
true: True,
squash: ↓T,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
cat-square-commutes: x_y1 o y1_z = x_y2 o y2_z
Lemmas referenced :
cat-functor_wf,
small-category_wf,
cat-ob_wf,
cat-arrow_wf,
functor-ob_wf,
equal_wf,
cat-comp_wf,
functor-arrow_wf,
set_wf,
mk-cat_wf,
cat-id_wf,
squash_wf,
true_wf,
functor-arrow-id,
cat-comp-ident2,
cat-comp-ident1,
iff_weakening_equal,
cat-comp-assoc,
functor-arrow-comp,
and_wf,
cat-square-commutes-comp,
cat-square-commutes_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
isect_memberEquality,
because_Cache,
productEquality,
applyEquality,
productElimination,
setEquality,
lambdaFormation,
setElimination,
rename,
lambdaEquality,
independent_isectElimination,
dependent_set_memberEquality,
independent_pairEquality,
natural_numberEquality,
imageElimination,
universeEquality,
dependent_functionElimination,
imageMemberEquality,
baseClosed,
independent_functionElimination,
independent_pairFormation,
equalityUniverse,
levelHypothesis,
applyLambdaEquality,
functionEquality,
hyp_replacement
Latex:
\mforall{}[A,B,C:SmallCategory]. \mforall{}[S:Functor(A;C)]. \mforall{}[T:Functor(B;C)]. ((S \mdownarrow{} T) \mmember{} SmallCategory)
Date html generated:
2017_10_05-AM-00_50_30
Last ObjectModification:
2017_07_28-AM-09_20_25
Theory : small!categories
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